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The problem is asking for the derivative \(\frac{d}{dx} f(g(x))\) at \(x = 6\), using the given table of values.

This requires the application of the **chain rule**, which states that:

\[
\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)
\]

We need to evaluate this at \(x = 6\).

### Step-by-step solution:

1. From the table, when \(x = 6\), we have:
   - \(g(6) = 6\)
   - \(g'(6) = 5\)

2. Now, we need \(f'(g(6))\). Since \(g(6) = 6\), we find \(f'(6)\) from the table:
   - \(f'(6) = 6\)

3. Applying the chain rule:
   \[
   \frac{d}{dx} f(g(x))\bigg|_{x=6} = f'(g(6)) \cdot g'(6) = f'(6) \cdot g'(6) = 6 \cdot 5 = 30
   \]

Thus, the derivative \(\frac{d}{dx} f(g(x))\) at \(x = 6\) is **30**.

Would you like further details or have any other questions?

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Here are 5 related questions:
1. What is the chain rule in calculus, and how is it applied?
2. How would the solution change if the function \(g(x)\) had different values at \(x = 6\)?
3. Can you compute \(\frac{d}{dx} g(f(x))\) at \(x = 6\) using the same table?
4. How would you find the second derivative of \(f(g(x))\) at \(x = 6\)?
5. How do the values of \(f'(x)\) and \(g'(x)\) impact the rate of change of the composition \(f(g(x))\)?

**Tip:** Always remember to apply the chain rule when dealing with composite functions!

Use the table of values to calculate the derivative of the function at the given point. Find d/dx [f(g(x))] at x = 6.

This problem demonstrates how to apply the chain rule to find the derivative of a composite function f(g(x)) at x = 6 using a table of values. The step-by-step solution uses key calculus concepts, such as the chain rule, to solve the problem.

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